EE313 Linear Systems and Signals Spring 2013

Initial conversion of content to PowerPointby Dr. Wade C. Schwartzkopf

Prof. Brian L. Evans

Dept. of Electrical and Computer Engineering

The University of Texas at Austin

Z-transforms

21 - 2

Z-transforms• For discrete-time systems, z-transforms play

the same role as Laplace transforms do in continuous-time systems

• As with the Laplace transform, we compute forward and inverse z-transforms by use of transforms pairs and properties

n

nznhzH )(

Bilateral Forward z-transform

R

n dzzzHj

nh 1 )( 2

1][

Bilateral Inverse z-transform

21 - 3

Z-transform Pairs• h[n] = [n]

Region of convergence: entire z-plane

• h[n] = [n-1]

Region of convergence: entire z-plane except z = 0

h[n-1] z-1 H(z)

1 )(0

0

n

n

n

n znznzH

11

1

1 1)(

zznznzHn

n

n

n

1 if 1

1

)(

00

z

a

za

z

aza

znuazH

n

n

n

nn

n

nn

• h[n] = an u[n]

Region of convergence: |z| > |a| which is the complement of a disk

21 - 4

Region of Convergence• Region of the complex z-

plane for which forward z-transform converges

Im{z}

Re{z}Entire plane

Im{z}

Re{z}Complement of a disk

Im{z}

Re{z}Disk

Im{z}

Re{z}

Intersection of a disk and complement of a disk

• Four possibilities (z=0 is a special case and may or may not be included)

21 - 5

Applying Z-transform Definition• Finite extent signal

Polynomial in z-1

• Five-tap delay lineImpulse response h[n]

Transfer function H(z)

h[n]

n1 2 3 4 5

12345

5

0

)(n

n

n

n znhznhzH

h[n] = n ( u[n] – u[n-6] )

54321 54320)( zzzzzzH

n = -1:6;h = n .* ( stepfun(n,0) - stepfun(n-6,0) );stem(n, h);

21 - 6

azza

nuaZ

n

for 1

11

BIBO Stability• Rule #1: Poles inside unit circle (causal signals)• Rule #2: Unit circle in region of convergence

Analogy in continuous-time: imaginary axis would be in region of convergence of Laplace transform

• Example:

BIBO stable if |a| < 1 by rule #1

BIBO stable if |z| > |a| includes unit circle; hence, |a| < 1 by rule #2

BIBO means Bounded-Input Bounded-Output

21 - 7

Inverse z-transform• Definition

• Yuk! Using the definition requires a contour integration in the complex z-planeUse Cauchy residue theorem (from complex analysis) OR

Use transform tables and transform pairs?

• Fortunately, we tend to be interested in only a few basic signals (pulse, step, etc.)Virtually all signals can be built from these basic signals

For common signals, z-transform pairs have been tabulated

R

n dzzzHj

nh 1 )( 2

1][

21 - 8

Example• Ratio of polynomial z-

domain functions• Divide through by the

highest power of z• Factor denominator

into first-order factors• Use partial fraction

decomposition to get first-order terms

21

23

12)(

2

2

zz

zzzX

21

21

21

23

1

21)(

zz

zzzX

11

21

121

1

21)(

zz

zzzX

12

1

10 1

21

1)(

z

A

z

ABzX

21 - 9

Example (con’t)• Find B0 by polynomial

division

• Express in terms of B0

• Solve for A1 and A2

15

23

2121

2

3

2

1

1

12

1212

z

zz

zzzz

11

1

121

1

512)(

zz

zzX

8

21

121

21

1

21

921

441

1

21

1

1

21

2

2

1

21

1

1

1

z

z

z

zzA

z

zzA

21 - 10

Example (con’t)• Express X(z) in terms of B0, A1, and A2

• Use table to obtain inverse z-transform

• With the unilateral z-transform, or the bilateral z-transform with region of convergence, the inverse z-transform is unique

11 1

8

21

1

92)(

zzzX

nununnxn

82

1 9 2

21 - 11

Z-transform Properties• Linearity

• Right shift (delay)

Second property used in solving difference equations

Second property derived in Appendix N of course reader by decomposing the left-hand side as follows:x[n-m] u[n] = x[n-m] (u[n] – u[n-m]) + x[n-m] u[n-m]

)()( 22112211 zXazXanxanxa

)( zXzmnumnx m

m

l

lmm zlxzzXznumnx1

)(

21 - 12

Z-transform Properties

)()( 21

21

21

21

21

2121

2121

zFzF

zrfzmf

zrfmf

zmnfmf

zmnfmf

mnfmfZnfnfZ

mnfmfnfnf

r

rm

m

m r

mr

m n

n

n

n

m

m

m

• Convolution definition

• Take z-transform

• Z-transform definition

• Interchange summation

• Substitute r = n - m

• Z-transform definition